| L(s) = 1 | + (1.22 + 0.708i)2-s + (−0.741 + 1.56i)3-s + (0.995 + 1.73i)4-s + (−1.92 + 0.339i)5-s + (−2.01 + 1.39i)6-s + (1.11 + 0.937i)7-s + (−0.0120 + 2.82i)8-s + (−1.90 − 2.32i)9-s + (−2.60 − 0.950i)10-s + (−0.757 − 0.133i)11-s + (−3.45 + 0.271i)12-s + (1.24 − 3.43i)13-s + (0.702 + 1.93i)14-s + (0.896 − 3.27i)15-s + (−2.01 + 3.45i)16-s + (2.40 + 4.17i)17-s + ⋯ |
| L(s) = 1 | + (0.865 + 0.501i)2-s + (−0.427 + 0.903i)3-s + (0.497 + 0.867i)4-s + (−0.862 + 0.152i)5-s + (−0.823 + 0.567i)6-s + (0.422 + 0.354i)7-s + (−0.00427 + 0.999i)8-s + (−0.633 − 0.773i)9-s + (−0.822 − 0.300i)10-s + (−0.228 − 0.0402i)11-s + (−0.996 + 0.0784i)12-s + (0.346 − 0.952i)13-s + (0.187 + 0.518i)14-s + (0.231 − 0.844i)15-s + (−0.504 + 0.863i)16-s + (0.584 + 1.01i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.506 - 0.862i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.506 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.756851 + 1.32165i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.756851 + 1.32165i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.22 - 0.708i)T \) |
| 3 | \( 1 + (0.741 - 1.56i)T \) |
| good | 5 | \( 1 + (1.92 - 0.339i)T + (4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (-1.11 - 0.937i)T + (1.21 + 6.89i)T^{2} \) |
| 11 | \( 1 + (0.757 + 0.133i)T + (10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (-1.24 + 3.43i)T + (-9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-2.40 - 4.17i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-6.24 - 3.60i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.35 + 1.13i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (2.25 + 6.19i)T + (-22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-6.71 + 5.63i)T + (5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (4.91 - 2.83i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.331 - 0.120i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-6.05 - 1.06i)T + (40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-1.39 - 1.16i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + 3.71iT - 53T^{2} \) |
| 59 | \( 1 + (11.1 - 1.96i)T + (55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-5.27 + 6.28i)T + (-10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-2.10 + 5.79i)T + (-51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (3.64 + 6.30i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.685 + 1.18i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (16.2 - 5.89i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-2.31 - 6.36i)T + (-63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (6.47 - 11.2i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.57 + 8.90i)T + (-91.1 - 33.1i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.43837078348681162582528465344, −11.75887556613687886309555798348, −11.01213365776742366760263390324, −9.897761520410245343293796917098, −8.297584756199236915761355482521, −7.73847029981315524990468660067, −6.07515834047406042873945599471, −5.34108424847366366236114566712, −4.09994139327543444293406059528, −3.19863436843086484178356455947,
1.19432835103335537097107920051, 3.02680779234152904790143950375, 4.55868435380061820198678404822, 5.49927752571179016352381086304, 6.98205298386358975935491267199, 7.51831435560046298051738515390, 9.054937561247540439275753347476, 10.54791469681303343766250561369, 11.59213633247357897449557451053, 11.78121705605445670822627260430
